Actuation Voltage For a Cantilever Switch in MEMS
Preamble
Following the successful 3D simulation of the process flow of cantilever based MEMS switches using VICTORY Process is presented in August 2005 Simulation Standard article “Process Flow Simulation and Manufacture”, a novel analytical method to predict the actuation voltage for such switches is presented here. This will help extend the utility of the process simulation to the prediction of the required voltage needed for cantilever switches and the circuit design required to drive such switches.
1. Introduction
When a system of conductors is charged, or subjected to some external electric potentials, the charge will entirely reside on the surfaces of the conductors and will distribute itself on each closed surface such as to make that surface equipotential and the field inside the conductor material zero. The charge on each surface will interact with itself and with other charges, which will result in net forces on each surface element. The net force on each surface element is normal to the surface, and given by the formula
(1)
where E is the electric field at the element, σ is the surface charge density there and δS is the area of the element. So, this force acts in a pressurelike manner, and can be expressed in the form of a stress vector
(2)
This stress is called Maxwell’s stress.
The field and the surface charge are related to each other thus
(3)
giving the formula for the Maxwell stress vector
(4)
where n is the outer unit normal on the surface.
The total force on each surface is calculated by integrating the stress (4) over the surface.
Since conductors are elastic bodies, in a stationary state the Maxwell stress will be balanced by mechanical stresses which will develop inside the conductors. These stresses will be the result of elastic deformation of the conductors. Our problem consists in finding this deformation as a function of the applied voltage.
2 Free Energy Formulation for a Thin Plate
A plate is thin if its thickness is small compared with its other two dimensions. For the case of small deformations (i.e. small in comparison with the length of the plate) considerable simplifications result, not least in the problem becoming essentially twodimensional.
The differential formulation of the problem is still quite complex. It involves the solution of an inhomogeneous biharmonic equation in 2D for the deformation (which is a partial differential equation of the fourth order, with the Maxwell stress as a load), with rather complex boundary conditions (see Landau & Lifshitz, eqns. (12.5)  (12.7)).
However, it is possible to formulate the Helmoltzlike free energy for the problem and then find a good approximation for the deformed shape which will minimize this energy. This free energy for the constant voltage situation is (Ljepojevic & Forbes 1995)
(5)
where C is the capacitance of the conductor, V the applied voltage and F_{elastic} the elastic energy. Placing the initial position of the plate in the horizontal xy plane, the elastic energy is given by (Landau & Lifshitz, eq. (11.6))
(6)
where E_{Y} is the Young modulus, σ_{P} the Poisson ratio, h the thickness of the plate and ζ the vertical displacement of the point initially in the horizontal plane.
3 Approximation to the Elastic Energy of a Cantilever with No Twisting
For a cantilever which is clamped along one edge (say the line y=0), and with no twisting assumed, the vertical displacement ζ is a function of x only, so eq. (6) reduces to
(7)
where W is the width of the plate and L its length.
As the cantilever bends (and provided this bending happens slowly and progressively, i.e. no violent vibration is present), the electrostatic force will increasingly become concentrated at the free end (i.e. near x = L), so the shape will resemble that of a cantilever loaded at one end. Therefore, as the amount of the maximum allowed bending (i.e. before it touches the other electrode) is quite small, we can approximate it with a quadratic polynomial
(8)
where ζ_{0} is the position of the clamped end, and a and b coefficients to be determined.
Choosing the coordinate system such that ζ_{0} = 0 and noting that for a cantilever dζ/dx = 0 at the clamped end (i.e. at x = 0, see Figure 1), we have
(9)
i.e. only one parameter (i.e. b) to be determined. With this, we have
(10)
so eq. (7) becomes
(11)
Figure 1. Cantiliver shape approximation for stress. 
4. Approximation to the Capacitance
The bending of the plate is very slight, so for the purpose of calculating the capacitance, we can approximate it with a linear function (see Figure 2).
Figure 2. Cantiliver shape approximation for capacitance. 
The capacitance is a function of the angle θ. The relationship between θ and b is obtained from
(12)
i.e.
(13)
Expanding the capacitance as a function of θ about the initial position of the plate (i.e. point O, see Figure 3a), we would have
(14)
In the region of very small θ, C(θ) is approximatelly an antisymmetric function of θ, so the second derivative there is zero. It follows that the neglected terms in eq. (14) are of the third order in θ. Regardless of the fact that the maximum angle θ_{c} = d_{0}/L is very small (typically L = 50μ and the distance between the plates d_{0} = 1μ, so θ_{c} = 0.02) the capacitance changes significantly by the time the free end touches the bottom plate. This indicates that either the first derivative is large or that higher derivatives are significant. Given that θ is very small throughout and given that the capacitance remains finite as the free end approaches the bottom plate we conclude that higher derivatives may become significant only in the very narrow range of θ approaching θ_{c}. So, the third order accurate eqno (14) should be a good approximation. However, this fomula needs the first derivative of C. We can obtain it numerically by solving the capacitance problem for several tilted positions of the lever and then differentiate numerically.

Figure 3. a) Expansion about point O; b) Expansion about point H. 
However, it is possible to estimate the change of the capacitance with θ without doing any further calculations. Namely, for every position of the cantilever, one can see it as tilted not about point O, but about point H in the middle of the lever (see Figure 3b). The advantage of doing this is that if the capacitance is represented by that of a plate tilted about the moving point H, the correction due to the tilting is of the second order in θ and can be neglected. This is due to the capacitance with tilting about point H being a symmetric function of θ, so the first derivative at H is zero. So, we can write
(15)
where O(θ2) represents the order of the neglected terms. Therefore we have
(16)
or expressed through b,
(17)
So, to obtain the capacitance of a plate tilted about the point O second order accurate in θ, we need to calculate the capacitance of the horizontal plate at the distance d_{0} − Lθ.
So far this is just circumventing the CLEVER’s present inability to treat tilted levers (CLEVER can easily treat the parallel ones). But it is actually much more than that. Noting the fact that d_{0} is much smaller than L and W, one may represent the functional dependence of the capacitance on the plate distance by that for a planparallel capacitor. So we can write equation (17) as:
(18)
where C_{0} is the capacitance at the intital position, i.e. at b_{0}.
So we see that all that is needed is the initial value of the capacitance only. We note that for the maximum value of θ (i.e. θ = θ_{c} and the corresponding value of b = b_{c} = d_{0}/L^{2}, when the free end touches the lower plate) the value of the denominator in eq. (18) is d_{0} − Lθ_{c} = d_{0}, so that our approximation gives for the capacitance at θ_{c} a value twice the original one, i.e. the change is quite large. The real change would in the very narrow region around θc be much larger (possibly there is a singularity), but outside this narrow region, i.e. during most of the lever travel, the formula (18) is probably quite good.
5. Calculation of the Actuation Voltage
The free energy can now be written as
(19)
The equilibrium condition is δF/δb = 0, giving
(20)
So, we have a cubic equation for b, with the voltage as a parameter. There are three cases: i) three distinct real solutions (all positive), b1 < b2 < b3; ii) three real solutions (all positive), but two coincide, b1 = b2 > b3; and iii) there are two complex solutions and one real solution b3 > 0, depending on the voltage (see Figure 4).
Figure 4. Graphs of and as functions of b and V. 
Solution b_{3} is unphysical in all three cases, as it is greater than b_{c}, which is the maximum value of b reachable. In the case (i), solution b_{1} is stable, whereas solution b_{2} is unstable. In the case (ii), the double root b_{1} = b_{2} is unstable. We are interested in the case (i) if we can adjust the voltage such that the smallest root b_{1} becomes equal to b_{c}, and if not then we consider the case (ii), the so called critical case.
It can be seen from Figure 4 that the case (i) is the first case encountered as we start increasing the voltage from zero. So, we will first encounter the smallest root which is the equilibrioum position of the lever bequilib = b1 ≤ bc. This equilibrium is stable, so we then need to finetune the voltage to V = Vc such that bequilib becomes equal to bc, i.e. that the lever just touches the lower plate, if possible. This would be the minimumum voltage that could operate the switch. Note that the critical case, when we have the double root situation, would then require a higher voltage, but then bcrit would be greater than bc so the equlibrium voltage Vc would suffice. If we cannot obtain the situation where the smallest root b1 equals bc, then we should consider the case (ii).
We can find the actuation voltage V_{a} in the following way. If we substitute b_{c} for b straight into eq. (20) and then solve for V, we will have one solution of the cubic for the voltage:
(21)
The other two roots with this voltage are
(22)
We see that the root b = b_{c} = d_{0}/L^{2} is between these two, i.e. b_{c} = b_{2}, so it is not the one we need. So we conclude that our solution is the case (ii), i.e. the critical case.
If we increase the voltage slightly, the roots b_{2} and b_{1} will move toward each other very quickly, to reach b_{crit}, because the crossing is already near a tangent situation. The critical solution b_{1} = b_{2} = bcrit is less than b_{c}, so there is no continuous solution to the closed position of the switch, but after reaching the unstable point bcrit, the lever would snap into that position, with some pressing force.
We could take Vc as a good approximation to the actuation voltage, but further analysis shows that b_{crit} = b_{c}, so the actuation voltage is
(23)
i.e. V_{a} = 1.09 V_{c}. Also, we can see that with this value of bcrit, the Eq. (18) for the capacitance is in the region of good approximation.
For design purposes, it is useful to have a rough feeling on how C_{0} depends on other geometrical parameters. One can write it as C_{0} = α_{C}WL/d_{0}, where α_{C} is a correction factor which remains approximatelly constant for small variations of parameters, so we can write Eq. (23) as
(24)
i.e.
(25)
We can see that the actuation voltage increases with d_{0} as ∼ d_{0}^{3/2}, and decreases with the length of the plate as ∼ L^{2}.
The dependence on the plate width W is contained in the factor α_{C}, and is therefore weak.
References
 Landau LD and Lifshitz EM, Theory of Elasticity, Pergamon Press, Oxford 1997.
 Ljepojevic NN and Forbes RG, Proc. Roy. Soc. A450, 177 (1995).